Theorem nnssn0s 28214

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28209	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4099	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3993	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3911   ⊆ wss 3914  {csn 4589   0_s c0s 27734  ℕ_0scnn0s 28206  ℕ_scnns 28207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-ss 3931  df-nns 28209

This theorem is referenced by:  nnssno  28215  nnn0s  28220  nnn0sd  28221  nnsgt0  28231